Let’s face it. anyone who has worked on Time Series Forecasting problems in the retail, logistics, e-commerce etc. would have definitely cursed that long tail which never behaves. The dreaded intermittent time series which makes the job of a forecaster difficult. This nuisance renders most of the standard forecasting techniques impractical, raises questions about the metrics, model selection, model ensembling, you name it. And to make things worse, there may be cases(like in the spare parts industry, where intermittent patterns appear is slow-moving but highly critical or high value items.

Notation

Y - i th element of the time series

n - The index of timeseries

i - The index of non-zero demand

Qi - The inter-demand interval, i.e. the gap between two non-zero demand.

Mi- The demand size at non-zero demand point.

Traditional Approach

Traditionally, there is a class of algorithms which take a slightly different path to forecasting the intermittent time series. This set of algorithms considered the intermittent demand in two parts — Demand Size and Inter-demand Interval — and modelled them separately.

Croston

Croston proposed to apply a single exponential smoothing seperately to both M and Q, as below:

After getting these estimates, final forecast,

And this is a one-step ahead forecast and if we have to extend to multiple timesteps, we are left with a flat forecast with the same value.

Croston(SBA)

Syntetos and Boylan, 2005, showed that Croston forecasting was biased on intermittent demand and proposed a correction with the β from inter demand interval estimation.

Croston(SBJ)

Shale, Boylan, and Johnston (2006) derived the expected bias when the arrival follow a Poisson process.

Croston Forecasting as Renewal Process

Renewal process is an arrival process in which the interarrival intervals are positive, independent and identically distributed (IID) random variables (rv’s). This formulation generalizes Poison process for arbitrary long times. Usually, in a Poisson process the inter-demand intervals are exponentially distributed. But renewal processes have an i.i.d. inter-demand time that have a finite mean.

Turkmen et al. 2019 casts Croston and its variants into a renewal process mold. The random variables, M and Q, both defined on positive integers fully define the Yn

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